Fixed Point Theorems And Applications Of Anti Mixed Monotone Operators And Integral Operators

Differential equation is an important subject in math and it is a very indispensabletool in people’s production practice. Mixed monotone operator theory is an active field ofdifferential equation and mixed monotone operator theory is extremely important to studythe nonlinear differential equation. Facts have proved that the in-depth study of mixedmonotone operator theory is of great significance. Because mixed monotone operatortheory has been widely used in aerospace science, technology, biochemical science,engineering, economics, ecology and so on many domains. When mixed monotoneoperator theory was first stated by GuoDaJun professors and V.Lakshmikantham professorin1987, mixed monotone operator theory caused the interest of many experts and scholars.From then on many good results were raised. However, unfortunately, the research of antimixed monotone operator theory is still in its infancy, so to discuss such problem is verynecessary.Differential equation plays a important role in the development of modern scienceand production practice. Because differential equation can describe the local problems ofphysics, biology, chemistry, engineering problems and so on. And to this kind of problemespecially positive solutions of research had been under concerned.In this paper, the existence and uniqueness theorems of solutions for the systems ofanti mixed monotone operators was discussed. And the last part of this paper is about theexistence of positive solutions of two-Point boundary value problems for second orderequations. This paper is divided into four parts:First, the purpose and meaning of the research of the mixed monotone operatorstheory and the boundary value problems for second differential equations, and the researchstatus and the main content at home and abroad. And the related concepts and theorem inthe research of the mixed monotone operators theory and the boundary value problems ofdifferential equation is also listed there.Second, by using cone and partial ordering theory and the skill of monotone iteration,the existence and uniqueness theorems of solutions for the systems of anti mixedmonotone operators which do not contain any continuity and compactness conditions areobtained.Third, by using cone and partial ordering theory and the skill of monotone iteration,the existence and uniqueness theorems of solutions for the systems of concave-convex anti -tmixed monotone operators which do not contain any continuity and compactnessconditions are obtained.Finally, some fixed point theorems of integral operators are given firstly by usingfixed point index theory. These results can be used to solve many problems. In this paper,applying these results, we study the existence of positive solutions for two-point boundaryvalue problems in Banach space is normal, which is essential for the technique used inmany papers.